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A120934 Least prime p such that the interval [p,p+log(p)] contains n primes. 2
2, 11, 457, 3251, 165701, 10526557, 495233351, 196039655873, 10687033762033, 79006533276941, 4313367040646743, 1740318019946551931 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Soundararajan states that, on average, there is one prime in the interval [k,k+log(k)] for any number k. Is there an upper limit to the number of primes in such an interval? Not if the prime k-tuple conjecture is true, in which case a(n) exists for all n. Note that a(n) > e^A008407(n). See A120935 for the largest prime in the interval.

a(n) begins a sequence of n primes whose prime pattern is one of the patterns in the n-th row of A186634. For example, the sequence of four consecutive primes beginning with 3251 is (3251, 3253, 3257, 3259), which has pattern (0, 2, 6, 8), which is in the 4th row of A186634.

LINKS

Table of n, a(n) for n=1..12.

K. Soundararajan, The distribution of prime numbers

Eric Weisstein's World of Mathematics, Prime k-Tuple Conjecture

FORMULA

This sequence grows superexponentially; a weak lower bound is a(n) >> (log n)^n. It seems that a(n) > n^n. - Charles R Greathouse IV, Apr 18 2012

EXAMPLE

a(2)=11 because p=11 is the first prime with log(p)>2 and 11+2 is prime.

MATHEMATICA

i=1; Table[While[p=Prime[i]; PrimePi[p+Log[p]]-PrimePi[p]+1< n, i++ ]; p, {n, 5}]

CROSSREFS

Cf. A120936 (number of primes in the interval [n, n+log(n)]), A020497.

Sequence in context: A012950 A012979 A013109 * A000886 A128855 A262547

Adjacent sequences:  A120931 A120932 A120933 * A120935 A120936 A120937

KEYWORD

hard,nonn,nice

AUTHOR

T. D. Noe, Jul 21 2006

EXTENSIONS

a(12) from Donovan Johnson, Apr 18 2012

STATUS

approved

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Last modified December 10 11:58 EST 2016. Contains 279001 sequences.